Beam polarization asymmetries for the p(gamma-->,K+)Lambda and p(gamma-->,K+)Sigma(0) reactions are measured for the first time for E(gamma)=1.5-2.4 GeV and 0.6<cos((theta(c.m.)(K+))<1.0 by using linearly polarized photons at the Laser-Electron-Photon facility at SPring-8 (LEPS). The observed asymmetries are positive and gradually increase with rising photon energy. The data are not consistent with theoretical predictions based on tree-level effective-Lagrangian approaches. Including the new results in the development of the models is, therefore, crucial for understanding the reaction mechanism and to test the presence of baryon resonances which are predicted in quark models but are thus far undiscovered.